Incidence matrices and line graphs of mixed graphs

TL;DR

This paper extends a fundamental theorem relating incidence and adjacency matrices from undirected graphs to mixed graphs, providing aligned definitions to generalize the line graph concept.

Contribution

It introduces aligned definitions for matrices and line graphs of mixed graphs, generalizing a key theorem from undirected graph theory.

Findings

Established a generalized theorem for mixed graphs

Provided aligned matrix definitions for mixed graphs

Extended line graph concepts to mixed graphs

Abstract

In the theory of line graphs of undirected graphs there exists an important theorem linking the incidence matrix of the root graph to the adjacency matrix of its line graph. For directed or mixed graphs, however, the exists no analogous result. The goal of this article is to present aligned definitions of the adjacency matrix, the incidence matrix and line graph of a mixed graph such that the mentioned theorem is valid for mixed graphs.